Higgs Bundles and Harmonic Maps Workshop
January 3-11, 2015

We will emphasize and study the role of equivariant harmonic maps as a crucial link between Higgs bundles, geometric structures and representation varieties. After preliminary talks on harmonic maps, Higgs bundles and the geometry of symmetric spaces, further topics will include: Corlette's theorem on the existence of equivariant harmonic maps, relationships between harmonic maps and integrable systems, and harmonic maps to singular spaces. Subsequent talks will be based on current developments in the theory of harmonic maps, Higgs bundles and geometric structures. In particular, applications to studying minimal surfaces in symmetric spaces, deformation spaces of OPERS, the geometry of Anosov representations with quasi-Fuchsian representations and Hitchin representations serving as illustrative examples, and asymptotic analysis of such deformation spaces. The overarching theme may be summarized as follows: how does one unpack the data of Higgs bundles, and via a careful study of harmonic maps, start to understand the geometry of surface group representations in a more explicit way.

Scientific Program

The workshop will consist of whiteboard talks by the participants on the following topics. Speakers will be alotted 2.5 hours per talk. Speakers will be asked to submit a 5-6 page summary, clicking on the each title below will lead you to that talks summary.

#### 1. Rob Maschal: Higgs bundles background

An outline of the nonabelian Hodge correspondence with focus on Higgs bundles, especially the Hitchin component and $SL(2,\mathbb{C})$.

##### Suggested literature:

• Hitchin's paper: Self duality equations on Riemann surfaces
• Hitchin's paper: Lie groups and Teichmüller space.
• Lectures by Steve Bradlow at GEAR junior retreat 2012.
• Lectures by Peter Gothen at Isaac Newton Institure for Mathematical Sciences.

• #### 2. Tengren Zhang: Geometry of Symmetric and Homogeneous spaces

Semisimple Lie groups: Cartan decompositions, Riemannian geometry, and boundaries of the associated symmetric spaces.

##### Suggested literature:

• Helgason's book: Differential geometry, Lie groups and symmetric spaces
• Chapter 1 of Burstall et al's book: Twistor theory for Riemannian Symmetric spaces.

• #### 3. Jérémy Toulisse: Existence theory for harmonic metrics.

Corlette's theorem

##### Suggested literature:

• Eels and Sampson paper: Harmonic mappings of Riemannian manifolds
• Simon Donaldson: Twisted harmonic maps and the self-duality equations
• Kevin Corlette: Canonical metrics on flat G-bundles
• François Labourie: Existence D'Applications Harmoniques Tordues à Valeurs Dans les Variétés à Courbure Négative.”

• #### 4. Semin Kim: Harmonic maps to $\mathbb{R}$-trees and Morgan Shalen compactification

##### Suggested literature:

• Mike Wolf's paper: On Realizing measured foliations via quadratic differentials of harmonic maps to R-trees
• Daskalopoulos, Dostoglu, and Wentworth paper: Character variety and harmonic maps to R-tree.”

• #### 5. Andrew Sanders: Background on Labourie's conjecture on existence and uniqueness of minimal surfaces for Hitchin representations.

##### Suggested literature:

• François Labourie's paper: Cross ratios, Anosov representations and the energy functional on Teichmuller space.
• David Baraglia's Thesis: G2 Geometries and integrable systems.

• #### 6. Marco Spinaci: Labourie's recent paper: Cyclic surfaces and Hitchin components in rank 2.

##### Suggested literature:

• François Labourie's paper: Cyclic surfaces and Hitchin components in rank 2

• #### 7. Qiongling Li: Background on harmonic maps to metric spaces and survey of Katzarkov, Noll, Pandit, and Simpson's paper: Harmonic maps to Buildings and Singular perturbation theory.

##### Suggested literature:

• Gromov and Schoen's paper: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one.
• Korevaar and Schoen's paper: Sobolev spaces and harmonic maps for metric space targets.
• Katzarkov, Noll, Pandit, and Simpson's paper: Harmonic maps to Buildings and Singular perturbation theory.

• #### 8. Jorge Acosta: OPERS and complex projective structures.

##### Suggested literature:

• Dalakov's Thesis: Higgs bundles and Opers
• Dumas's survey: Complex projective structures
• Wentworth's notes: Higgs bundles and local systems on Riemann surfaces.

• #### 9. Brian Collier: The relationship between Integrable systems and Harmonic maps with emphasis on the Toda lattice.

##### Suggested literature:

• Martin Guest's book: Harmonic Maps, Loop Groups, and Integrable systems
• Aspects of mathematics book: Harmonic Maps and Integrable systems (selected chapters)
• Bolton, Pedit and Woodwards paper: Minimal surfaces and the affine Toda model.

• #### 10. Andy Huang: The relationship between Integrable systems and Harmonic maps with emphasis on the Toda lattice.

##### Suggested literature:

• Martin Guest's book: Harmonic Maps, Loop Groups, and Integrable systems
• Aspects of mathematics book: Harmonic Maps and Integrable systems (selected chapters)
• Bolton, Pedit and Woodwards paper: Minimal surfaces and the affine Toda model.

• #### 11. Brice Loustau: Minimal surfaces in $\mathbb{H}^3$ and quasi-Fuchsian representations.

Minimal surfaces as equivariant harmonic maps, Taubes moduli space of minimal hyperbolic germs, explicit examples of corresponding $SL(2,\mathbb{C})$ Higgs bundles.

##### Suggested literature:

• Taubes paper: Moduli space of minimal hyperbolic germs
• Donaldson's paper: Moment maps in differential geometry
• Thomas Hodge's paper: Hyper-Kahler geometry and Teichmüler space.

• #### 12. Jakob Blaavand: Exposition of Ends of the moduli space of Higgs bundles by Mazzeo, Swoboda, Weiss, Witt.

##### Suggested literature:

• Mazzeo, Swoboda, Weiss, Witt's paper: Ends of the moduli space of Higgs bundles

• #### 13. Laura Fredrickson: Survey of Taubes's paper $PSL(2,\mathbb{C})$-connections with $L^2$ bounds on curvature.

##### Suggested literature:

• Taubes's paper: $PSL(2,\mathbb{C})$-connections with $L^2$ bounds on curvature.

• #### 14. Daniele Alessandrini: Branched hyperbolic surfaces and nonmaximal $SL(2,\mathbb{R})$ representations/Higgs bundles.

##### Suggested literature:

• Hitchin's paper: The self-duality equations on a Riemann surface
• Goldman's paper: Higgs bundles and geometric structures on surfaces.

• Participants

Organizers: Brian Collier, Qiongling Li, Andrew Sanders
Schedule and Notes

 9am 2:30pm 8pm Saturday Robert Maschal Sunday Tengren Zhang Jeremy Toulisse Monday Semin Kim Andy Sanders Tuesday Marco Spinaci Qiongling Li Wednesday Jorge Acosta Thursday Laura Fredrickson Brice Loustau Friday Jakob Blaavand Andy Huang Saturday Brian Collier Daniele Alessandrini
Logistics

#### Location:

The workshop will take place January 3-11, 2015 in a large cabin in the mountains of North Carolina near Asheville.

#### Transportation:

Transportation between Asheville airport and the cabin will be provided on Saturday, January 3, and Sunday, January 11. Please email your itinerary to the organizers as soon as it becomes available.

#### Funding:

Full funding for the workshop is provided by the GEAR network. Invited participants will be reimbursed for travel (see your email). Please email your receipts to the organizers prior to the workshop or bring printed copies with you.
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